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Chun, C. and Chun, D. and Moss, T. and Noble, Steven (2018) The
e-Exchange Basis Graph and Matroid connectedness. Discrete Mathematics
342 (3), pp. 723-725. ISSN 0012-365X.
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CONNECTEDNESS
CAROLYN CHUN, DEBORAH CHUN, TYLER MOSS, AND STEVEN NOBLE
Abstract. LetM be a matroid ande∈E(M). Thee-exchange basis
graph ofM has vertices labeled by bases of M, and two vertices are adjacent when the bases labeling them have symmetric difference{e, x}
for somex∈E(M). In this paper we show that a connected matroid is exactly a matroid with the property that for every elemente∈E(M), thee-exchange basis graph is connected.
1. Preliminaries
Terminology will follow Oxley [6]. LetM be a matroid. The set of bases of M is denoted by B(M). In the basis graph GB(M), the vertices are
labeled by the the elements ofB(M). Let B1, B2 ∈ B(M). The verticesB1 and B2 are adjacent in GB(M) exactly when |B14B2|= 2.
It seems that interest in these structures for graphic matroids started in the mid 1960s [1] when they were called tree graphs. This name persisted even though Carlos Holzmann and Frank Harary studied them for general matroids [2]. Stephen Maurer completed the most in-depth treatment of the subject and called them basis graphs. His doctoral thesis and two resulting articles [3] and [4] were published in the early 1970s. More recently, they have been called basis exchange graphs or bases exchange graphs. This terminology has been attributed to Jack Edmonds [5]. While each name has particular advantages, we will refer to the structures by the shortest accepted name and call them basis graphs.
It is well known that the basis graph of a matroid is connected for any matroid. This is an easy result of the basis exchange axiom, and this result is listed as part of Theorem 2.1 in [3].
Lemma 1.1. Let M be a matroid. Then GB(M) is connected.
It is easy to see that the basis graph of a matroid is isomorphic to the that of its dual.
Lemma 1.2. Let M be a matroid. Then GB(M∗) ∼= GB(M). Further,
GB(M∗) is found by relabeling each vertex of GB(M) by its complement in
E(M).
Suppose a matroid M has element e. The e-exchange basis graph of
M, which we will denote GB,e(M), is a subgraph of GB(M). The vertex
2 CAROLYN CHUN, DEBORAH CHUN, TYLER MOSS, AND STEVEN NOBLE
B1, B2 ∈ B(M) exactly when B14B2 = {e, f} for some f ∈ E(M). It is easy to see that this graph is bipartite with one part containing all the bases containing e. The next lemma is related to Lemma 1.2.
Lemma 1.3. Let M be a matroid with e ∈ E(M). Then GB,e(M∗) ∼=
GB,e(M). Further, GB,e(M∗) is found by relabeling each vertex of GB,e(M)
by its complement in E(M).
Now consider the case whereeis a loop. Theneis avoided by every basis, and the following is an easy result.
Lemma 1.4. The graph GB,e(M) has no edges if and only if e is a loop or
coloop.
The main part of our result has previously been established in the tech-nical report [5], where machinery is developed showing that GB,e(M) is connected for every elementeif and only ifM is a connected matroid. The same result follows from elementary matroid operations, and this paper gives a shorter proof.
2. Main Result
While we have already seen that the basis graph is connected for every matroid, we see from Lemma 1.4 that there are matroids for which an e -exchange basis graph is not connected. In fact, we characterized the graphs having no edges, and this lemma points to the larger theorem that the connectedness of the e-exchange basis graphs of M is related to the con-nectedness of the underlying matroid. Our main result is that a matroidM
is connected exactly when for every elemente∈E(M), the graph GB,e(M) is connected.
Theorem 2.1. LetM be a matroid with at least two elements. The following are equivalent.
(1) If e∈E(M), then GB,e(M) is connected and has at least one edge.
(2) If e, f ∈E(M) are distinct, then there is a basis B ∈ B(M) so that B4{e, f} ∈ B(M).
(3) M is connected.
Proof. First, we show that (2) and (3) are equivalent. A matroid M is connected exactly when for every pair of elements {e, f} ∈ E(M) there is a circuit C containing {e, f}. This occurs exactly when there is a basis B
containingf but note, such thatf is containted in the fundamental circuit of B∪e. This occurs if and only ifB4 {e, f} is also a basis ofM.
Now, we assume (1) and prove that (2) follows. Let e, f ∈E(M). Let Ve be the set of vertices of GB(M) labeled by bases containing e. Let Vf be the set of vertices ofGB(M) labeled by bases containing f.
Suppose, for the sake of contradiction, that Ve ⊆ Vf. Then every basis containing e also contains f. As GB,f(M) is connected and bipartite with
GB,e(M) is connected and bipartite withVe as one part, this graph contains an edge from vertex A to some vertex A0 of Ve. Observe that A0 contains
e and f. As bases are equicardinal, {e, f} is a proper subset ofA4A0 and
|A4A0|> 2, a contradiction. Therefore, the supposition is false and there is a basis, Be ofM so thatBe∈Ve−Vf. Symmetrically, there is a basisBf of M so thatBf ∈Vf −Ve.
Again, relying on (1), we know GB,f(M) is connected, so there is a path from Be to Bf. Somewhere on this path, there is an edge connecting two bases with symmetric difference{e, f}. Thus (1) implies (2).
Finally, we demonstrate that (2) implies (1). Therefore, assume (2). For the sake of contradiction, we assume that there is an element e∈E(M) so thatGB,e(M) is not connected.
By assumption, we know for anyf ∈E(M)−ethat{e, f}is the symmetric difference of the labels of their endvertices. Thus we conclude that the second part of (1) holds: GB,e(M) has at least one edge.
By our assumption, GB,e(M) is comprised of n ≥ 2 components,
χ1, χ2, . . . , χn. By Lemma 1.1, GB(M) is connected. So M has distinct
bases Bi and Bj so that Bi ∈ χi and Bj ∈ χj and Bi4Bj ={ei, ej}. We assume ei ∈Bi and ej ∈Bj.
If e∈ {ei, ej}, this would indicate an edge in GB,e(M) between different
components, a contradiction. Thus eithere∈Bi∩Bj oreavoids both bases. Relying on Lemma 1.3, up to duality, we may assume the former is true.
The set H = cl(Bi −e) is a hyperplane. Notice ei ∈ H. Label the complementary cocircuit, C∗ =E−H. For f ∈C∗−e, the set Bi4{e, f} is a basis ofM. The elementej is not inC∗. OtherwiseBi4{e, ej} ∈ B(M) making Bi4{e, ej} adjacent to Bi and Bj in GB,e(M), a contradiction to their residing in different components. Becauseej ∈H, clearlyH = cl(Bj−
e). And so forf ∈C∗−e, the set Bj4{e, f} is a basis of M.
In order to complete this proof, we will need to examine the structure of these two components in detail. It will be helpful to label the edges by the symmetric difference of the labels of the vertices.
Consider the subgraph of GB,e(M) induced by Bi and its neighbors. As
GB,e(M) is bipartite, this graph is a star. The edges are labelled by{e, f}for each elementf inC∗−e. Exactly the same is true for the subgraph induced by Bj and its neighbors. Moreover the subgraph induced by Bj and its neighbors may be obtained from that induced by Bi and its neighbors by replacing Bi with Bj and replacing each occurrence of ei with ej in the vertex labels.
For each f ∈ C∗ −e, the preceding argument may be repeated with
4 CAROLYN CHUN, DEBORAH CHUN, TYLER MOSS, AND STEVEN NOBLE
may be applied to every pair of components of GB,e(M), so the same edge
labels appear in each component. Since ei appears in every basis labeling a vertex of the component χi, the set {e, ei} does not appear as an edge label in the first component. Thus this set labels no edge inGB,e(M). This
contradicts (2), proving our claim.
References
[1] R. L. Cummins, Hamiltonian circuits in tree graphs,IEEE Transactions of Circuit Theory 13(1966), 82–90.
[2] F. Harary and C. Holzmann, On the tree graph of a matroid,SIAM Journal of Applied Mathematics 22(1972), 187–193.
[3] S. Maurer, Matroid basis graphs. I,J. Combin. Theory Ser. B 14(1973), 216–240. [4] S. Maurer, Matroid basis graphs. II,J. Combin. Theory Ser. B 15(1973), 121–145. [5] M. Mihail and M. Sudan, Connectivity Properties of Matroids,EECS Department,
University of California, Berkeley Technical Report UCB/CSD-92-662(1991). [6] J. Oxley,Matroid Theory, Second edition, Oxford University Press, New York, 2011.
(C. Chun)Department of Mathematics, United States Naval Academy, An-napolis, MD, USA
E-mail address: [email protected]
(D. Chun)Department of Mathematics, West Virginia University Institute of Technology, Beckley WV, USA
E-mail address: [email protected]
(T. Moss)Department of Mathematics, West Virginia University Institute of Technology, Beckley WV, USA
E-mail address: [email protected]
(S. Noble)Department of Economics, Mathematics and Statistics, Birkbeck,
University of London, London, United Kingdom
